Problem 34 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |x-a|

Step-by-Step Solution

Answer

\( (a-d, a+d) \)

Step 1: Understanding Absolute Value Inequality

The inequality \(|x-a| < d\) represents a range of values for \(x\) that are within a distance \(d\) from \(a\). This implies that the expression \(|x-a|\) can be rewritten as the compound inequality: \[ -d < x-a < d \]

Step 2: Isolating x

To solve for \(x\), we need to isolate \(x\) in the compound inequality. We'll do this by adding \(a\) to all parts of the inequality.\[ -d + a < x < d + a \]

Step 3: Writing in Interval Notation

The solution \(-d + a < x < d + a\) can be expressed in interval notation as:\((a-d, a+d)\). This interval includes all values of \(x\) greater than \(a-d\) and less than \(a+d\).

Key Concepts

Absolute Value Inequality

An absolute value inequality involves an expression of the form \(|x-a| < d\). Here, the absolute value expression \(|x-a|\) measures the distance of \(x\) from \(a\). \(d\) is the maximum allowable distance, and the inequality states that \(x\) must be within this distance from \(a\).

• For any number inside the absolute value, say \(b\), \(|b|\) represents how far \(b\) is from zero, regardless of direction.
• In the inequality \(|x-a| < d\), \(x\) can take any value that makes the distance from \(a\) less than \(d\).

Understanding this concept helps us rewrite the absolute value inequality into a more manageable form, specifically a compound inequality, and solve it step by step.

Interval Notation

Interval notation is a shorthand method of expressing the set of solutions for an inequality. For the inequality \-d < x-a < d\, which is derived from the absolute value inequality, we rephrase it into the simpler interval notation form.

• This form uses parentheses or brackets to indicate whether endpoints are included (closed intervals) or not (open intervals).
• For the inequality \-d+a < x < d+a\, the solution in interval notation will be \( (a-d, a+d) \).
• The parentheses indicate that the endpoints \(a-d\) and \(a+d\) are not included in the solution set.

Interval notation is efficient and visually intuitive, showing the solution on a number line, letting us see at a glance which numbers satisfy the given conditions.

Compound Inequality

A compound inequality is an equation with two inequalities joined together. In our context, the absolute value inequality \(|x-a| < d\) becomes the compound inequality \(-d < x-a < d\).

• This implies that the value of \(x\) must satisfy both conditions of the inequality simultaneously.
• To solve it, follow the steps within each part of the inequality while maintaining the relations linked by the inequality sign.
• Adding or subtracting the same number from all three parts of the inequality is a common technique used to isolate the variable \(x\).

In this scenario, by isolating \(x\) through algebraic manipulation, we arrive at the solution as \-d+a < x < d+a\, and express it seamlessly in interval notation as \( (a-d, a+d) \). Understanding compound inequalities facilitates solving complex problems by simplifying them into smaller, manageable parts.